Anisotropic Shubin operators and eigenfunctions expansions in Gelfand-Shilov spaces

2016/09/20 by Marco Cappiello, Todor Gramchev, Cappiello, Marco +5 · 4 citations
Mathematics · #Advanced Mathematical Physics Problems #Spectral Theory in Mathematical Physics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.1609.06214

Abstract

We derive new results on the characterization of Gelfand--Shilov spaces S^μ_ν(\Rⁿ), μ,ν>0, μ+ν≥ 1 by Gevrey estimates of the L² norms of iterates of (m,k) anisotropic globally elliptic Shubin (or Γ) type operators, (-Δ)^m/2 +| x |^k with m,k∈ 2\N being a model operator, and on the decay of the Fourier coefficients in the related eigenfunction expansions. Similar results are obtained for the spaces Σ^μ_ν(\Rⁿ), μ,ν>0, μ+ν> 1, cf. \eqrefGSdef. In contrast to the symmetric case μ= ν and k=m (classical Shubin operators) we encounter resonance type phenomena involving the ratio κ:=μ/ν; namely we obtain a characterization of S^μ_ν(\Rⁿ) and Σ^μ_ν(\Rⁿ) in the case μ=kt/(k+m), ν= mt/(k+m), t ≥ 1, that is, when κ=k/m ∈ \Q.

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• Global hypoellipticity for a class of time-periodic operators on asymptotically Euclidean manifolds
• Sequence space representations of Beurling-Björck spaces via Gabor frames and Wilson bases

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